Israel Journal of Mathematics

, Volume 207, Issue 2, pp 617–651 | Cite as

Structural connections between a forcing class and its modal logic

  • Joel David Hamkins
  • George Leibman
  • Benedikt Löwe
Article

Abstract

Every definable forcing class Γ gives rise to a corresponding forcing modality \({\square _\Gamma }\) where \({\square _{\Gamma \varphi }}\) means that ϕ is true in all Γ extensions, and the valid principles of Γ forcing are the modal assertions that are valid for this forcing interpretation. For example, [10] shows that if ZFC is consistent, then the ZFC-provably valid principles of the class of all forcing are precisely the assertions of the modal theory S4.2. In this article, we prove similarly that the provably valid principles of collapse forcing, Cohen forcing and other classes are in each case exactly S4.3; the provably valid principles of c.c.c. forcing, proper forcing, and others are each contained within S4.3 and do not contain S4.2; the provably valid principles of countably closed forcing, CH-preserving forcing and others are each exactly S4.2; and the provably valid principles of ω1-preserving forcing are contained within S4.tBA. All these results arise from general structural connections we have identified between a forcing class and the modal logic of forcing to which it gives rise.

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Copyright information

© Hebrew University of Jerusalem 2015

Authors and Affiliations

  • Joel David Hamkins
    • 1
    • 2
  • George Leibman
    • 3
  • Benedikt Löwe
    • 4
    • 5
  1. 1.Mathematics ProgramThe Graduate Center of The City University of New YorkNew YorkUSA
  2. 2.Department of MathematicsThe College of Staten Island of CUNYStaten IslandUSA
  3. 3.Department of Mathematics and Computer Science, Bronx Community CollegeThe City University of New YorkBronxUSA
  4. 4.Institute for Logic, Language and ComputationUniversiteit van AmsterdamAmsterdamThe Netherlands
  5. 5.Fachbereich MathematikUniversität HamburgHamburgGermany

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